Generalization of hypergeometric type differential equation

In summary, differential equations of the form shown can be solved using Mellin transforms if certain restrictions are placed on the variables involved. These restrictions include σ(s) being at most a 2nd-degree polynomial, τ(s) being at most a 1st-degree polynomial, and λ being a constant. However, if λ is a 3rd-degree polynomial, the method may not be applicable and it is unclear how to solve the transformed equation. It is also uncertain if there are analytical solutions for the equation if λ is a 3rd-degree polynomial.
  • #1
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I am aware that hypergeometric type differential equations of the type:

upload_2018-6-18_19-12-53.png


can be solved e.g. by means of Mellin transforms when σ(s) is at most a 2nd-degree polynomial and τ(s) is at most 1st-degree, and λ is a constant. I'm trying to reproduce the method for the case where λ is not constant, but a 3rd-degree polynomial, and I'm not sure how to solve Mellin-transformed version of the equation above if λ is a polynomial and not a constant, if indeed it is solvable analytically.

Does the equation above have analytical solutions if λ(s) is a 3rd-degree polynomial, and if so, how do I arrive at them?
 

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  • #2
I played around with equations "similar" to these in graduate school. I would just point out (forgive if this is obvious) if ##\sigma(s)## and ##\tau(s)## are restricted to polynomials of second and first order then the equation viewed as an operator on ##y(s)## sends polynomials of order ##n## into polynomials of order ##n## which I believe is what makes them "special". Letting ##\lambda## be a polynomial in ##s## will break this property.
 

What is a generalization of hypergeometric type differential equation?

A generalization of hypergeometric type differential equation is a type of differential equation that can be expressed in terms of the hypergeometric function. It has the form y'' + P(x)y' + Q(x)y = 0, where P(x) and Q(x) are functions of x.

What is the significance of hypergeometric type differential equations in science?

Hypergeometric type differential equations have many applications in science, including in physics, engineering, and statistics. They can be used to model various physical phenomena and to solve problems involving probability and statistics.

How is a generalization of hypergeometric type differential equation solved?

A generalization of hypergeometric type differential equation can be solved using various methods, including the Frobenius method and the Laplace transform. The solution can also be expressed in terms of the hypergeometric function or its derivatives.

What are some real-world examples of hypergeometric type differential equations?

Hypergeometric type differential equations can be used to model various physical systems, such as the motion of particles in a gravitational field, the flow of fluids, and the behavior of electrical circuits. They can also be applied in statistics to solve problems involving binomial distributions and hypergeometric distributions.

Are there any limitations to using hypergeometric type differential equations?

Hypergeometric type differential equations have certain limitations, such as being applicable only to linear systems and not being able to model nonlinear systems. They also require initial or boundary conditions to find a unique solution.

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